
\section{Model}

\label{sec:model}

\begin{table}[t!]
  \caption{Summary of Major Notation (External parameters and system
    variables are separated)}
  \begin{centering}
    \begin{tabular}{|c|l|}
      \hline
      \em External Parameter & \em Description\tabularnewline
      \hline
      \hline
     N & Number of users per one macro BS\tabularnewline
      \hline
      $\gamma,\bar{\gamma}, \theta$ & user type, max. of user type, price sensitivity\tabularnewline
      \hline
      $\beta$ & fraction of a femto BS's coverage\tabularnewline
      \hline
      $\delta_{o}$ & probability that a user is outside \tabularnewline
      \hline
      $C_M, C_F$ & capacities of macro and femto BSs  \tabularnewline
      \hline
$C_C, C_O$ & capacities of closed and open femto BSs  \tabularnewline
      \hline
$\eta$ &  femto capacity reduction factor\tabularnewline
      \hline
      $c_f$ & maintenance cost of a femto BS  \tabularnewline
      \hline \hline
   $q_{o}$ & fraction of coverage of all open femto BSs \tabularnewline
     \hline
    $ U_j, \Phi_j$ &expected utility and service fee of service type $j$\tabularnewline
\hline
    $ \bm{\alpha} = (\alpha_m , \alpha_o, \alpha_c)$ &user subscription ratios\tabularnewline
    \hline
    $ R,S,W$ &revenue, user surplus, social welfare\tabularnewline
    \hline
\end{tabular}\label{notations}
    \par\end{centering}
\end{table}


\subsection{System Model}


Consider a wireless network consisting of macro and femto BSs, where $N$
users/macro-cells are served by a monopoly operator.  We assume a simple
model of BSs, that is, macro and femto BSs provide the fixed capacities
$C_M$ and $\eta C_F$, where $C_F$ is the ``pure'' capacity of a femto BS
and $\eta \in (0,1]$ is an interference factor. The value of $\eta$
depends on spectrum sharing and femto open policy, whose details will be
discussed in Section~\ref{sec:interference}.
%~\footnote{The femto capacity is different for 
%different service model due to interference, which will be discussed in the later subsection.},$ respectively. 
%We assume a simple model of BSs that macro, open-femto, and closed-femto
%BSs provide the fixed capacities $C_M,$ $C_{O},$ and $C_C,$
%respectively. 




% \textbf{The capacity
% of femto BSs can be differentiated according to whether the femto BS
% is open or closed, which will be explained in the following
% subsection. For the distinction of capacities, $C_O$ and $C_C$ denote
% the capacities of open femtocells and closed femtocells,
% respectively.}

% \textbf{The femto BSs generate significant interference to their
% nearby macro users when the femto BSs use the same spectrum band with
% the macro BSs. Especially in closed femtocells, since the accessing of
% guests is not allowed, the nearby users have to be served from the
% macrocell even though the signal from the femto BS is much stronger,
% which generates coverage hole where the users cannot be served.
% To circumvent the interference issue, some operators assign
% independent resources with macrocells to femtocells.  However, using
% separate carriers induces significant loss on spectral efficiency
% because some part of resources are wasted.}

% \textbf{In order to solve the spectral inefficiency of the separate using, partially
% shared approaches are suggested. For example, when the operator has
% multiple carriers, called ``partially shared carriers'' in
% \cite{HC09DO}, femtocells use only part of the available carriers, so
% that macro users can avoid the interference from the femtocells. In
% ``partially shared carriers'',  the macrocells can use the whole frequency
% resources, because the carriers assigned to the closed femto BSs is
% available for the users who are far from femtocells. Therefore, the performance
% degradation of macrocells is not significant. Similarly, in OFDMA networks,
% the network can manage the frequency resources as ``partially shared
% carriers'' by resource management algorithms with just one spectrum
% band \cite{AC10}. From now, for the sake of clarity, {\em partially
% sharing} denotes the spectrum sharing schemes.}

% \textbf{Under {\em partially sharing}, not only macro BSs but also
% open femto BSs can use whole frequency resources, because the
% interference issues is removed by handoff where the users suffered by
% the signal of the femto BS change the serving BS to the open femto
% BS. Thus, the capacity of closed femto BSs becomes smaller than the
% capacity of open femto BSs due to the spectrum loss. Let $F_F$ be the
% fraction of the available carriers for partially using. Then, under
% {\em open-to-all}, the capacities of femtocells become $C_O = C_F$ and
% $C_C =F_F \cdot C_F$. However, note that $C_O = F_F \cdot C_F$, under
% {\em open-to-femto}. It is because open femto BSs also have to use a
% part of carriers where they do not allow the access of {\em
% mobile-only} users.}



% In
% Section~\ref{sec:spectrum_sharing}, we will introduce the actual
% capacities of femto BSs (based on $C_F$), which differ when they are
% open or closed.


Users are always guaranteed to be under the coverage of
a macro BS, but not of a femto BS. We also assume that femtocell
equipment is identical and that the coverage size of a femto BS is the
fraction $\beta$ of that of a macro BS. We do not consider the handover
effects on the users, and assume that each user already has a backhaul
connection based on residential Internet service, e.g., DSL (Digital
Subscriber Line).  We adopt this simple model to purely focus on the
economic aspects of the system and to enable meaningful analysis.

Note that the instantaneous BS capacities and the volume of users'
delivered data may largely depend on factors such as % bandwidth sharing
% policy (sharing or splitting the frequency bands between macro and
% femto BSs),
resource allocation mechanisms, and other factors (e.g.,
power control policy, channel conditions, user's distance to BS).
However, simplifying the model does not overly change the key messages
and insights because we focus on economic benefits and pricing, which
typically take effect over a longer time-scale.
% \textbf{In practical, to apply closed femto services, due to interference
% issue, the channel of
% femtocells should be separated with the channel of
% macrocells\cite{HC09DO}. Thus, we consider separate channel model for
% femtocell networks. Therefore, we could ignore the capacity reduction
% due to interference.}


% for the averaged user behaviors (e.g.,
% average volume of data for users' utility)

% which typically requires the average are enough to track the delivered
% data volume averaged over a longer time scale than the above-mentioned
% factors.

% Then, the impact of bandwidth sharing policy can be simply
% modeled by the actual values of $C_M$ and $C_F.$ As we will discuss
% later, our analysis and key messages are not sensitive to those values.

% \subsection{Spectrum Sharing and Femtocell Capacities}
% \label{sec:spectrum_sharing}

% \textbf{The capacity
% of femto BSs can be differentiated according to whether the femto BS
% is open or closed, which will be explained in the following
% subsection. For the distinction of capacities, $C_O$ and $C_C$ denote
% the capacities of open femtocells and closed femtocells,
% respectively.}

% \textbf{The femto BSs generate significant interference to their
% nearby macro users when the femto BSs use the same spectrum band with
% the macro BSs. Especially in closed femtocells, since the accessing of
% guests is not allowed, the nearby users have to be served from the
% macrocell even though the signal from the femto BS is much stronger,
% which generates coverage hole where the users cannot be served.
% To circumvent the interference issue, some operators assign
% independent resources with macrocells to femtocells.  However, using
% separate carriers induces significant loss on spectral efficiency
% because some part of resources are wasted.}

% \textbf{In order to solve the spectral inefficiency of the separate using, partially
% shared approaches are suggested. For example, when the operator has
% multiple carriers, called ``partially shared carriers'' in
% \cite{HC09DO}, femtocells use only part of the available carriers, so
% that macro users can avoid the interference from the femtocells. In
% ``partially shared carriers'',  the macrocells can use the whole frequency
% resources, because the carriers assigned to the closed femto BSs is
% available for the users who are far from femtocells. Therefore, the performance
% degradation of macrocells is not significant. Similarly, in OFDMA networks,
% the network can manage the frequency resources as ``partially shared
% carriers'' by resource management algorithms with just one spectrum
% band \cite{AC10}. From now, for the sake of clarity, {\em partially
% sharing} denotes the spectrum sharing schemes.}

% \textbf{Under {\em partially sharing}, not only macro BSs but also
% open femto BSs can use whole frequency resources, because the
% interference issues is removed by handoff where the users suffered by
% the signal of the femto BS change the serving BS to the open femto
% BS. Thus, the capacity of closed femto BSs becomes smaller than the
% capacity of open femto BSs due to the spectrum loss. Let $F_F$ be the
% fraction of the available carriers for partially using. Then, under
% {\em open-to-all}, the capacities of femtocells become $C_O = C_F$ and
% $C_C =F_F \cdot C_F$. However, note that $C_O = F_F \cdot C_F$, under
% {\em open-to-femto}. It is because open femto BSs also have to use a
% part of carriers where they do not allow the access of {\em
% mobile-only} users.}

\subsection{Monopoly Operator and Services}

As mentioned in Section~\ref{sec:intro}, the operator provides three
services, namely, \emph{mobile-only}, \emph{mobile+open femto}, and
\emph{mobile+closed femto}, and two femto open-policies: {\em
  open-to-all} and {\em open-to-femto}. We use
  $\{m,o,c\}$ to indicate these service types.


The operator charges $\phi_l^L(x)$ for generating traffic rate $x$ in
the BS type $L$ to the user subscribing to service $l \in \{m,o,c\}.$
The index $L$ in the charging function aims to show the
dependence of charging by the serving BS type. We consider two types
of tariffs: {\em flat pricing} and {\em partial volume pricing}.  We
use $\{M,O,C\}$ to index to macro, open-femto, and closed-femto BSs.

In flat pricing, users' payments are constant regardless of data
usage, that is,
% $$\phi_j({\bm x)} = p_j, \,\,\, j \in \{m,o,c\} .$$
for any BS type $L \in \{M, O, C \},$
\begin{equation}
\phi_l^L(x) = p_l, \,\,\, l \in \{m,o,c\} ,
\end{equation}
where $p_l$ is the constant charge for service $l \in \{m,o,c\}.$

In partial volume pricing, users pay $p_v^M$ per unit data rate when
they are served by macro BSs, whereas they pay a fixed service fee $p_m,
p_o, p_c$ for using femto BSs. This hybrid setup is motivated by the
practical reasons that low-cost femto BSs may not be appropriately
equipped for complex per-data operations. Recall that operators do not
use volume pricing for femto services \cite{femto_forum}. Thus, for all
$l \in \{m,o,c \}$, the pricing structure is represented as follows:
\begin{eqnarray}
  \phi_l^M(x) &=& p_v^M x + p_l, \\
  \phi_l^L(x) &=& p_l, \ L \in \{ O,C\}, \\
  p_m &=&0,
\end{eqnarray}
where $p_m =0$ is due to the fact that (pure) volume pricing is applied
when {\em mobile-only} users access macro BSs but cannot access femto
BSs.

\subsection{Capacity and Interference Model}
\label{sec:interference}

The interference between macro and femto BSs depends on a spectrum
sharing policy and service types. Under the separate carriers in which
femto and macro BS do not share the spectrum, we can assume that the
macro and femto capacities are fixed to be $C_M$ and $C_F$. However,
when they share the carriers, e.g., {\em partially shared carriers}
\cite{HC09DO}, the ``actual'' capacities differ depending on the type of the
femto BSs.  In the case of closed femto, macro users around the closed
femto BS can interfere the femto BS, possibly degrading the femto
capacity. However, if a femto cell is open, mobile-only users can
handoff to the open femto BS. To reflect this, we introduce the
interference factor $\eta$ and model a femto BS' capacity as:
\begin{eqnarray*}
  C_C &=& \eta C_F \qquad  \text{closed femto BS}, \cr
  C_O &=& C_F \qquad \ \: \text{open femto BS},
  % C_O &=& C_F \qquad \ \: \text{open and {\em open-to-all}}, \cr
  % C_O &=& \eta C_F \qquad \text{open and {\em open-to-femto}},
\end{eqnarray*}
where $C_F$ is again the ``pure'' femto BS capacity, $C_C$ and $C_O$ are
``actual'' capacities of the femto BS under closed and under open-femto
policies, respectively, $\eta \in (0,1].$ We assume that the macro
capacity $C_M$ is not affected, which can be justified from the fact
that interference mitigation scheme can be employed, e.g., {\em
  partially shared carriers} \cite{HC09DO}.


%An example for interference
%mitigation among femto and macro BSs is a spectrum sharing policy,
%called {\em partially shared carriers} \cite{HC09DO}, where macro BSs
%use the entire spectrum bands but femto BSs can only utilize a part of
%the bands.




%This assumption is
%based on separate carriers model\cite{HC09DO}, since we ignore
%the interference between macro and femto BSs. Note that, under shared
%carrier model, open femto policy gives more system capacity than closed
%femto policy.
%The femto BSs may generate non-negligible interference to
%nearby macro users, when femto and macro BSs share same spectrum bands.

%We separate the capacities of open and closed femto BSs, because the
%generated interference depends on when femto BSs are open or closed.
%Thus, we model a femto BS' capacity as:
%\begin{eqnarray*}
%  C_C &=& \eta C_F \qquad  \text{closed femto BS}, \cr
%  C_O &=& C_F \qquad \ \: \text{open femto BS},
%  % C_O &=& C_F \qquad \ \: \text{open and {\em open-to-all}}, \cr
%  % C_O &=& \eta C_F \qquad \text{open and {\em open-to-femto}},
%\end{eqnarray*}
%where $C_F$ is the ``pure'' femto BS capacity when no coupling (e.g.,
%spectrum sharing to handle interference) between macro and femto BSs
%is considered, and $\eta$ is the {\em femto capacity reduction
%  factor}, i.e., the fraction of reduced capacity due to interference
%mitigation schemes, $0< \eta \le 1.$ An example for interference
%mitigation among femto and macro BSs is a spectrum sharing policy,
%called {\em partially shared carriers} \cite{HC09DO}, where macro BSs
%use the entire spectrum bands but femto BSs can only utilize a part of
%the bands.


%\subsection{Economic Metrics: Users}
\subsection{Users }
%%%%%%%%%%%%%\
% REMOVED  by MO
%%%%%%%%%%%%%%%%%

%\textbf{Nowadays, with introducing smart phones and tablets, data
%communications become more important than voice connections in
%cellular networks. In data communications, users want high speed data
%transfer\cite{ACM06MH}. Thus, in this paper, we assume that the payoffs of users are
%determined by the data rate.

% In addition, we also assume that
% users always subscribe to a backbone service so that femto BSs could connect
% to the network. We ignore the economic effect that cellular networks
% replace xDSL services and just focus on the policy of femtocells.
%}

To model the behavior of a user, we adopt an
iso-elastic utility function~\footnote{An utility function $U(x)$
is said to be {\em iso-elastic} if for all $k>0,$ $U(kx) = f(k) U(x) +
g(k)$ for some functions $f(k),g(k) >0.$}
$u(x;\gamma)$, given by
\begin{eqnarray}
u(x;\gamma) &=& \gamma x^\theta,
\end{eqnarray}
where $x$
is traffic, $\gamma$ is a user type value, and $\theta$ is an elasticity
parameter. The utility function is an increasing concave function of
traffic volume $x$, but with a decreasing marginal payoff. The user type
parameter $\gamma$ is introduced to model different willingness to pay.
As the higher $\gamma$ users have the more payoff for the same
data-rate, they can afford more to subscribe to a service. We assume
$\gamma$ is uniformly distributed between $[0,\gamma_{max}],$
$\gamma_{max} >0.$ The
parameter $\theta$ is closely related to {\em elasticity of demand},
that is, the
percent change of demand to the percent change of price. A higher
value of
elasticity means more changes in demand to the price change. It is known
that the elasticity for the given utility function $u(x;\gamma)$ is
$\frac{1}{1-\theta}$.  The iso-elastic function for data traffic is used
in \cite{DL00PQ,MITRA01,HUANG09}.

% {\em iso-elastic} functions are widely
%used for utility functions in economics \cite{DL00PQ}. For example, concave
%functions is a kind of {\em iso-elastic} function. In this work,
%because the increment of payoff of users becomes smooth as the data
%rate increases, a concave function is suitable for the payoff function
%in data networks. The utility function is given by:
%\begin{equation} u(x;\gamma) = \gamma x^\theta,\end{equation}
%where $\gamma$ is a user type value, which is assumed to be uniformly
%distributed over the interval $[0,\bar{\gamma}]$ for some
%$\bar{\gamma}>0$. The higher $\gamma$ user has the more payoff for the
%same data-rate, which means that the higher $\gamma$ user can afford
%more to subscribe to a service.  Thus, $\gamma$ is a parameter for
%willingness to pay. In addition, in the utility function, $\theta \in
%[0,1]$ is elasticity. As $\theta \rightarrow 1$ the utility function
%becomes linear, while it approaches a step function as $\theta
%\rightarrow 0$.

As the service rates are dissimilar for macro, open and closed
femto BSs, we introduce {\em expected} utility and {\em expected}
service fee functions for service type $l$, $U_l ,$ and $\Phi_{l}$, as follows:
\begin{eqnarray}
 U_l({\bm{x}};\gamma) &=& \bexpect{ u(x^L;\gamma) }= \gamma \sum_{L \in
  \{M,O,C \}} (x^L)^\theta \pi_l^L, \\
\Phi_{l}(\bm{x}) &=& \bexpect{ \phi^L(x^L) } = \sum_{L \in  \{M,O,C \}} \phi_{l}^L(x^L)\pi_l^L.
\end{eqnarray}
where $\pi_l^L$ is  the fraction of time or probability that users of service type $l$ use a type $L$ BS to
get a service, and $\bm{x}=( x^M ,x^O ,x^C )$ is a vector that
represents the traffic volume generated in each type of BS.


Then, the net-utility $\tilde{U}_{l}$ of service type $l$ is given by
\begin{equation} \tilde{U}_{l}({\bm{x}};\gamma) = U_l({\bm{x}};\gamma) - \Phi_{l}({\bm x}),\,\,\, l\in \{m,o,c\}. \end{equation}


Users move and connect to different types of BSs over time. Users
achieve different data rates, which also depends on the service
type. Under our system model, when there are $n$ \emph{open-femto} users, the
fraction of area $q_o$ covered by the open femto BSs is given by:
\begin{eqnarray}
  \label{eq:qzero}
  q_{o} \triangleq 1- (1-\beta )^n.
\end{eqnarray}
Users' average mobility statistics are assumed to be equal.  This is
denoted by
$\delta_i$, which is the probability of being ``inside,'' where $\delta_o = 1
- \delta_i.$ To users of femto services, $\delta_i$ corresponds to the
fraction of time that they are under the coverage of their own femto BSs.  The
\emph{mobile-only} users rely on macro BSs even when they are
inside because of the absence of their own femto BSs. We ignore the
  possibility that the \emph{mobile-only} users utilize neighboring
  femto BSs when they are inside for simplicity. When users are outside, they
can access either a macro or an open-femto BS.  They access an
open-femto BS with a probability $q_o$ or a macro BS with a probability
$1-q_o$.

Table \ref{tbl:pi} shows $(\pi_l^L: l \in \{m,o,c\}, L \in \{M,O,C \})$
under different open policies.  Under the {\em open-to-femto} policy, \emph{mobile-only}
users cannot access open femto BSs and can only access macro BSs, as
shown on the first line. \emph{Open-femto} users access open femto BSs
with a probability $\delta_{o}q_{o}+\delta_{i}$ and macro BSs with
probability $\delta_{o} (1-q_{o})$.  The \emph{closed-femto} user case is shown
in a similar manner. Under the {\em open-to-all policy}, even \emph{mobile-only}
users can access the open femto BSs.

\begin{table}[t!]
   \caption{probability $\pi_l^L$ for open-to-femto  and open-to-all policies}
   \begin{centering}
     \tabcolsep=1.6mm
     \begin{tabular}{|c|c|c|c||c|c|c|}
       \hline
& \multicolumn{3}{c||}{\em Open-to-femto} &
       \multicolumn{3}{c|}{\em Open-to-all} \cr \hline & M & O & C & M & O & C\tabularnewline
       \hline
       m & 1& 0 & 0 & $\delta_{o} (1-q_{o}) + \delta_{i}$ & $\delta_{o}q_{o}$ & 0\tabularnewline
      \hline
       o & $\delta_{o} (1-q_{o})$ & $\delta_{o}q_{o}+\delta_{i}$ & 0 & $\delta_{o} (1-q_{o})$ & $\delta_{o}q_{o}+\delta_{i}$ & 0 \tabularnewline
      \hline
       c & $\delta_{o}(1-q_{o})$ & $\delta_{o}q_{o}$ & $\delta_{i}$ & $\delta_{o}(1-q_{o})$ & $\delta_{o}q_{o}$ & $\delta_{i}$\tabularnewline
      \hline
\end{tabular}\label{tbl:pi}
     \par\end{centering}
\end{table}


\subsection{Operators and Regulators}
According to the user type $\gamma$ and charging schemes, a user selects
a service type and decides on the data demand. Let $\bm{\alpha} =
(\alpha_l: l \in \{m,o,c \})$ be the vector of user fractions
subscribing to each service type $l.$ The service type and traffic
rate vector of the user type $\gamma$ are denoted by $l^{*}(\gamma)$ and $\bm{x}(\gamma),$
respectively. Thus, operator revenue $(R)$, social
welfare $(W)$, and user surplus $(S)$ are computed as
\begin{eqnarray}
R&=&\int\Phi_{l^{*}(\gamma)}({\bm
  x}(\gamma))Nd\gamma-(\alpha_{o}+\alpha_{c})Nc_{f},   \label{eq:revenue} \\
W&=&\int U_{l^{*}(\gamma)}({\bm
  x}(\gamma);\gamma)Nd\gamma-(\alpha_{o}+\alpha_{c})Nc_{f},   \label{eq:welfare}\\
S&=&\int\tilde{U}_{l^{*}(\gamma)}({\bm x}(\gamma);\gamma)Nd\gamma=W-R  \label{eq:surplus},
\end{eqnarray}
respectively, where $c_f$ is the cost of a femto BS for the service provider.

% \textbf{It seems that the operator pay all of cost from femtocells in
% \eqref{eq:revenue}. However, this model could be also regarded as
% users partially impose the cost or even cover all of femto costs. Let
% assume that $\Phi_{j}(\bm{x})$ is total payment of users who subscribe
% to $j \in \{ m,o,c \}$ which includes service fee and maintenance cost $c_f$
% for a femto BS deployed in their house. Then, users who subscribe to
% $j \in \{ o,c \}$ pay $\Phi_{j}(\bm{x}) - c_f$ to the
% operator. However, still, the operator's revenue, the social
% welfare, and the user surplus are represented as in
% \eqref{eq:revenue}, \eqref{eq:welfare}, and \eqref{eq:surplus}, respectively.}



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